Transformation Groups

, Volume 13, Issue 3–4, pp 819–837 | Cite as

Irregular and Singular Loci of Commuting Varieties

  • Vladimir L. Popov


Let \( {\user1{\mathcal{C}}} \) be the commuting variety of the Lie algebra \( \mathfrak{g} \) of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let \( {\user1{\mathcal{C}}}^{{{\text{sing}}}} \) be the singular locus of \( {\user1{\mathcal{C}}} \) and let \( {\user1{\mathcal{C}}}^{{{\text{irr}}}} \) be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) \( {\user1{\mathcal{C}}}^{{{\text{sing}}}} \) is a nonempty subset of \( {\user1{\mathcal{C}}}^{{{\text{irr}}}} \); (b) \( {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} \) where the maximum is taken over all simple ideals \( \mathfrak{a} \) of \( \mathfrak{g} \) and \( l{\left( \mathfrak{a} \right)} \) is the “lacety” of \( \mathfrak{a} \); and (c) if \( \mathfrak{t} \) is a Cartan subalgebra of \( \mathfrak{g} \) and \( \alpha \in \mathfrak{t}^{*} \) root of \( \mathfrak{g} \) with respect to \( \mathfrak{t} \), then \( \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} \) is an irreducible component of \( {\user1{\mathcal{C}}}^{{{\text{irr}}}} \) of codimension 4 in \( {\user1{\mathcal{C}}} \). This yields the bound \( {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} \) and, in particular, \( {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 \). The latter may be regarded as an evidence in favor of the known longstanding conjecture that \( {\user1{\mathcal{C}}} \) is always normal. We also prove that the algebraic variety \( {\user1{\mathcal{C}}} \) is rational.

Key words and phrases

Reductive Lie algebra commuting variety decomposition class irregular element singular point semisimple and nilpotent elements 

AMS classiffication

14M99 14L30 14R20 14L24 17B45 


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Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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